Our model (REF ) is generated by a wave equation on \(\mathbf {H}_n\)  (see, for example, [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]} and reference therein) coupled with a heat equation on \(\mathbf {H}_n\)  (see, for example, [9]}, [10]} and reference therein). However, this coupling adopted the lower-order terms \((\theta (t,\eta ),-\partial _tu(t,\eta ))^{\mathrm {T}}\)  will completely change some qualitative properties of solutions, which behave those neither for wave equations nor for heat equations on the Heisenberg group because we will observe regularity-loss type decay properties (see Theorem REF  later). Although some approximations and asymptotic profiles for the wave-heat coupled system were known in the Euclidean space, we should remark that these results have been obtained in [11]} only for the one-dimensional case. For this reason, we are interested in higher-dimensional cases, i.e.
